Multimodal Linguistic Inference

In this paper we compare grammatical inference in the context of simple and of mixed Lambek systems. Simple Lambek systems are obtained by taking the logic of residuation for a family of multiplicative connectives =; ; n, together with a package of structural postulates characterizing the resource management properties of the connective. Diierent choices for Associativity and Commutativity yield the familiar logics NL, L, NLP, LP. Semantically, a simple Lambek system is a unimodal logic: the connectives get a Kripke style interpretation in terms of a single ternary accessibility relation modeling the notion of linguistic composition for each individual system. The simple systems each have their virtues in linguistic analysis. But none of them in isolation provides a basis for a full theory of grammar. In the second part of the paper, we consider two types of mixed Lambek systems. The rst type is obtained by combining a number of unimodal systems into one multimodal logic. The combined multimodal logic is set up in such a way that the individual resource management properties of the constituting logics are preserved. But the inferential capacity of the mixed logic is greater than the sum of its component parts through the addition of interaction postulates, together with the corresponding interpretive constraints on frames, regulating the communication between the component logics. The second type of mixed system is obtained by generalizing the residuation scheme for binary connectives to families of n-ary connectives, and by putting together families of diierent arities in one logic. We focus on residuation for unary connectives, hence on mixed (2,3) frames. The unary connectives play the role of control devices, both with respect to the static aspects of linguistic structure, and the dynamic aspects of putting this structure together. We prove a number of elementary logical results for unary families of residuated connectives and their combination with binary families, and situate existing proposals for`structural modalities' within a more general framework. in this paper prepare the ground for the sections on multiplicative connectives in a chapter on categorial grammar for the Handbook of Logic and Language (Van Benthem and ter Meulen (eds), Elsevier, to appear). I thank the editors of the Handbook and the audiences at these workshops for comments. I have greatly beneeted from discussion, and joint work, with Natasha Kurtonina, Glyn Morrill and Dick Oehrle. All errors are my own.